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\title{\vspace{-4cm}\textbf{河北师范大学数学分析真题}}
\author{杨泽天}
\date{\today}
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\begin{document}
\date{}
\maketitle
\section*{2019年数学分析}
\begin{problem}
    求极限
    $$ 
    \displaystyle\lim_{n\to \infty } \frac{\left[ (n+1)(n+2)\cdots (n+n) \right]^\frac{1}{n}}{n}.
    $$
\end{problem}

\begin{problem}
    证明 $\displaystyle F(x)=\int_0^{+\infty} \frac{\cos t}{t^x}\d t $ 在 $[1,+\infty )$ 上连续，可微. 
\end{problem}

\begin{problem}
    证明 $\displaystyle \sum_{n=1}^{+\infty }a_n \cos x$ 一致收敛的充分必要条件是 $\displaystyle \sum_{n=1}^{+\infty }a_n$ 收敛.  
\end{problem}

\begin{problem}
    $f(x)$ 二阶连续可导， $\displaystyle \lim_{x\to 0}\frac{f(x)}{x}=0,f''(0)=4$,求 $\displaystyle \lim_{x\to 0} \left( 1+\frac{f(x)}{x} \right)^\frac{1}{x}.$  
\end{problem}

\begin{problem}
    曲面 $\Sigma$ 为 $z=0,z=b,x^2+y^2=a^2$ 外侧，求 
    $$ 
        \varoiint\limits_\Sigma x^3\d y\d z + x^2y\d x\d z+x^2z\d x\d y.
    $$
\end{problem}
\begin{problem}
    数分课本 P163.8
\end{problem}

\begin{problem}
    $\displaystyle z=z(x,y),F(x+\frac{z}{y},y+\frac{z}{x})$ 证明 $\displaystyle x \frac{\d z}{\d x}+y\frac{\d z}{\d y}=z-xy.$ 
\end{problem}

\begin{problem}
    $f(x),g(x)$ 连续，$g(x)>0$ ，证明： $\exists \xi \in \left[ a,b \right]$ 使得
    $$ 
        \int_a^b f(x)g(x)\d x = f(\xi)\int_a^b g(x)\d x.
    $$
\end{problem}
\begin{problem}
    求 $ 
       \displaystyle \sum \frac{n^2}{n!}x^n
    $
    的和函数.
\end{problem}
\end{document}